The Pythagorean Theorem - Educational Outreach

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Figure 2.3 is another original, visual proofattributed to Pythagoras. Modern mathematicians wouldsay that this proof is more ‘elegant’ in that the samedeductive message is conveyed using one less triangle. Eventoday, ‘elegance’ in proof is measured in terms of logicalconciseness coupled with the amount insight provided bythe conciseness. Without any further explanation on mypart, the reader is invited to engage in the mental deductivegymnastics needed to derive the sum-of-squares equalityfrom the diagram below.Figure 2.3: An Alternate Visual Proof by PythagorasNeither of Pythagoras’ two visual proofs requires theuse of an algebraic language as we know it. Algebra in itsmodern form as a precise language of numericalquantification wasn’t fully developed until the Renaissance.The branch of mathematics that utilizes algebra to facilitatethe understanding and development of geometric conceptsis known as analytic geometry. Analytic geometry allows fora deductive elegance unobtainable by the use of visualgeometry alone. Figure 2.4 is the square-within-the-square(as first fashioned by Pythagoras) where the length of eachtriangular side is algebraically annotated just one time.30
cabcFigure 2.4: Annotated Square within a SquareThe proof to be shown is called a dissection proof due to thefact that the larger square has been dissected into fivesmaller pieces. In all dissection proofs, our arbitrary righttriangle, shown on the left, is at least one of the pieces. Oneof the two keys leading to a successful dissection proof isthe writing of the total area in two different algebraic ways:as a singular unit and as the sum of the areas associatedwith the individual pieces. The other key is the need toutilize each critical right triangle dimension—a, b, c—atleast once in writing the two expressions for area. Once thetwo expressions are written, algebraic simplification willlead (hopefully) to the Pythagorean Theorem. Let us startour proof. The first step is to form the two expressions forarea.1 : AAAbigsquarebigsquarebigsquare A c ( a b)littlesquare2 4(122 4 ( Aab)&onetriangle) The second step is to equate these expressions andalgebraically simplify.2 : ( a b)aa22 2ab b b2 cset22 c22 c 4(212ab) 2ab31
Page 1: The PythagoreanTheoremCrown Jewel o
Page 4: The Pythagorean TheoremCrown Jewel
Page 7 and 8: Table of ContentsList of Tables and
Page 9 and 10: List of Tables and FiguresTablesNum
Page 11 and 12: Figures…continuedNumber and Title
Page 13 and 14: List of Proofs and DevelopmentsSect
Page 15 and 16: PrefaceThe Pythagorean Theorem has
Page 17 and 18: 1 : 2x2x2****22 13x 7 13x 7 0 : a
Page 19 and 20: 1) Consider the Squares“If it was
Page 21 and 22: s090045 0ss45450450090Figure 1.3: O
Page 23 and 24: Notice that the right-triangle prop
Page 25 and 26: The middle square (minus the donut
Page 27 and 28: Our proof in Chapter 1 has been by
Page 29 and 30: 2) Four Thousand Years of Discovery
Page 31: The proof Pythagoras is thought to
Page 35 and 36: Figure 2.6 is the diagram for a sec
Page 37 and 38: 1x:b2:Area(CBA)3:Area( ABC)Area(ABC
Page 39 and 40: Euclid’s proof of the Pythagorean
Page 41 and 42: First we establish that the two tri
Page 43 and 44: 1 2y b b: y b c c2:A3:A4:Aunshaded
Page 45 and 46: XB'A090 CBFigure 2.12: Euclid’s C
Page 47 and 48: 2.3) Liu Hui Packs the SquaresLiu H
Page 49 and 50: One of my favorites was Boxel, a ga
Page 51 and 52: 2.4) Kurrah Transforms the Bride’
Page 53 and 54: Note: as is the occasional custom i
Page 55 and 56: 2.5) Bhaskara Unleashes the Power o
Page 57 and 58: 2.6) Leonardo da Vinci’s Magnific
Page 59 and 60: In Figure 2.24, we enlarge Step 5 a
Page 61 and 62: But, this is precisely the nature o
Page 63 and 64: 2.8) Henry Perigal’s TombstoneHen
Page 65 and 66: In the century following Perigal, b
Page 67 and 68: Returning to Henry Perigal, Figure
Page 69 and 70: Figure 2.32 is President Garfield
Page 71 and 72: Even though the Cartesian coordinat
Page 73 and 74: B 2← γ →α y βx C-xC 2Figure
Page 75 and 76: Being polynomial in form, the funct
Page 77 and 78: Thus, the critical point ( C 2 ,0)i
Page 79 and 80: Sincexcpwas chosen on an arbitrary
Page 81 and 82: ∂F/∂x = ∂F/∂y = 0x(C - x) 2
Page 83 and 84:
Define a new function21 A2 A3 2xcp
Page 85 and 86:
2.11) Shear, Shape, and AreaOur las
Page 87 and 88:
Step 1: Cut the big square into two
Page 89 and 90:
In Table 2.3, we briefly summarize
Page 91 and 92:
We can formally state this similari
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What we will do in Section 3.2 is p
Page 95 and 96:
There are three Pythagorean-triple
Page 97 and 98:
Thus, if we multiply any given prim
Page 99 and 100:
The last equality shows that the in
Page 101 and 102:
A Pythagorean Quartet is a set of f
Page 103 and 104:
3.5) Pythagoras and the Three Means
Page 105 and 106:
yxy MHxab2ab2 y x FC y ab2ab 2ab y
Page 107 and 108:
Hero was the founder of the Higher
Page 109 and 110:
5 : Solve for area using the formul
Page 111 and 112:
4 : Construct the parallelogram wit
Page 113 and 114:
We already have introduced Thabit i
Page 115 and 116:
Some historians believe Stewart’s
Page 117 and 118:
In pre-computer days, both all four
Page 119 and 120:
One example of a periodic process i
Page 121 and 122:
The six trigonometric functions are
Page 123 and 124:
The following four relationships ar
Page 125 and 126:
As with any set of identities, the
Page 127 and 128:
Figure 3.17 can be used as a jumpin
Page 129 and 130:
The remaining two pillars are the L
Page 131 and 132:
a2Law of Cosines c2 b2 2bccos()The
Page 133 and 134:
An example of a Diophantine equatio
Page 136 and 137:
However, Euler’s Conjecture did n
Page 138 and 139:
4) Pearls of Fun and WonderPearls o
Page 140 and 141:
We have achieved five-digit accurac
Page 142 and 143:
140212213116)74370116)(74370(370)74
Page 144 and 145:
X X X X O X X O X XO X X O O OO X O
Page 146 and 147:
4.3) Earth, Moon, Sun, and StarsIn
Page 148 and 149:
The four-step solution follows. All
Page 150 and 151:
Figure 4.9 depicts Eratosthenes’
Page 152 and 153:
---High above the earth, we see the
Page 154 and 155:
The commonly accepted value is abou
Page 156 and 157:
1BC : tan( BAC)BABC BAtan(BAC)0BC
Page 158 and 159:
BD ACtan( 21 )Even for the closest
Page 160 and 161:
3w 2h s w 5 s w 0.38196s&5 1s
Page 162 and 163:
Figure 4.16: Triangular Phi---The h
Page 164 and 165:
What remains to be done is to devel
Page 166 and 167:
3 : h32 4 h222 2 24C 2 h33162 2 2
Page 168 and 169:
Epilogue: The Crown and the Jewels
Page 170 and 171:
Not only has it endured, but the Py
Page 172 and 173:
A] Greek AlphabetGREEK LETTERUpper
Page 174 and 175:
C] Geometric FoundationsThe Paralle
Page 176 and 177:
8. Congruent Triangles: Two triangl
Page 178 and 179:
3. Rectangle: A bh : P 2b 2h, b &
Page 180 and 181:
Recreational Mathematics11. Pickove
Page 182 and 183:
GGolden RatioDefined and algebraica
Page 184 and 185:
Puzzles (cont)Pythagorean Magic Squ
Page 186 and 187:
Pythagorean Theorem ProofsAlgebraic
Page 188:
Trigonometry (cont)Eratosthenes mea
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The Pythagorean Theorem - Educational Outreach

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